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Calculating the surface area of a sphere is a fundamental concept in geometry that is often visualized in comparing celestial bodies like Earth and the Moon. The formula to compute the surface area of a sphere is given by \(4 \pi r^{2}\), where \(r\) stands for the radius of the sphere.

When we apply this formula to celestial bodies, we first ensure that their radii are expressed in the same units. For Earth and the Moon, when their radii are in meters, we can then square these values, multiply by four, and then by \(\pi\), the mathematical constant representing the ratio of a circle's circumference to its diameter. The result will be the surface area in square meters for each, allowing us to compare the two by taking the ratio of the Earth's surface area to that of the Moon. This comparison reveals how much larger or smaller one is compared to the other in terms of surface coverage.

Volume calculation for spheres like Earth and the Moon involves using another key geometric formula, \(\frac{4}{3} \pi r^{3}\). Here, we are interested in finding out how much three-dimensional space these celestial bodies occupy.

To calculate the volume, we cube the radius of the sphere, which is a critical step as it greatly amplifies the difference between sizes of celestial bodies due to the cubic power. We then multiply the cubed radius by \({\pi}\) and \(\frac{4}{3}\). This will yield the volume in cubic meters. The resulting values, when the radii of Earth and Moon are given in the same unit (meters), can then be divided to find their ratio. This ratio is a significant comparison that can be used to understand the scale of Earth compared to the Moon.

Unit conversion is an essential skill in many scientific calculations, ensuring consistent units across all measurements for accurate comparison. In our context, the Moon's radius was initially given in centimeters (cm), but as the standard practice in physics dictates using meters (m), we must convert the radius from centimeters to meters. The simple conversion is based on the fact that 1 meter equals 100 centimeters.

To do this conversion, we divide the number of centimeters by 100, thus moving the decimal point two places to the left. For example, \(1.74 \times 10^{8}\) cm becomes \(1.74 \times 10^{6}\) m. After conversion to meters, the radii of both Earth and Moon can then be used directly in our surface area and volume calculations.

Understanding the properties of a sphere is pivotal when studying celestial bodies like Earth and the Moon. A sphere is a perfectly round geometrical object in three-dimensional space that is the same distance from its center to its surface at all points. This characteristic imparts spheres with unique properties such as symmetry and an absence of edges or vertices.

Two critical properties come in the form of formulas for surface area \(4 \pi r^{2}\) and volume \(\frac{4}{3} \pi r^{3}\), revealing how a change in radius can significantly affect the size and space a sphere occupies. Notably, these properties allow us to compute and compare metrics such as size and mass when we look at spherical objects in the universe, which can be approximated as spheres for these types of calculations.